Ziv-Zakai Bound Research Articles

Deep learning based 2D-DOA estimation using L-shaped arrays

Direction-of-arrival (DOA) estimation problems arise in many applications such as wireless communication and localization. Recently, a number of deep learning (DL) based methods have been studied for one dimensional (1D) DOA estimation with relatively fewer studies for 2D-DOA estimation. In this study, we propose a low-complexity DL based method to estimate both elevation and azimuth DOAs of sources along with their pairing. To this end, first a classification neural network is proposed to estimate both elevation and azimuth DOAs of multiple sources. Next, a residual classification network is introduced to estimate the pairing between the estimated DOAs. In particular, we consider two perpendicular linear arrays which are located on z-axis and x-axis, respectively. The first proposed network uses the Sample Covariance Matrix (SCM) of the former array to estimate elevation angles and utilizes the latter array to estimate the corresponding azimuth DOAs. The second network, is responsible for estimating the pairing between the estimated elevation and azimuth DOA sets. Numerical simulations demonstrate the enhanced performance of our proposed 2D-DOA estimator scheme, surpassing existing 1D and 2D deep learning (DL) methods. Notably, our approach closely approaches the Ziv–Zakai bound (ZZB), particularly in low signal-to-noise ratio (SNR) and low-angle-difference scenarios, even in the presence of multiple highly correlated signals. Moreover, our complexity analysis validates the superiority of the proposed method.

Trade-Off Between Positioning and Communication for Millimeter Wave Systems With Ziv-Zakai Bound

In this paper, we investigate the trade-off between positioning and communication for an integrated positioning and communication (IPAC) millimeter wave system. First, in terms of the positioning in the IPAC system, the Cramér-Rao bound (CRB) is commonly used as a performance metric. Unfortunately, the CRB is only tight in a certain region for the high signal-to-noise ratio (SNR). To compensate for this deficiency, we derive the Ziv-Zakai bound (ZZB) for the IPAC system by exploiting the a priori delay information extracted from both the time delay parameter and the channel amplitude attenuation. Further, we derive the expected CRB (ECRB) and the weighted CRB (WCRB) of the system for comparisons. Second, we analyze the trade-off between positioning and communication for this IPAC system based on the derived ZZB. Specifically, we aim to optimize the power allocation to maximize the achievable data rate subject to the ZZB and total transmit power constraints. Numerical results show that the ZZB provides a tighter and more reasonable bound for the minimum mean square error (MMSE) estimator over the wide range of SNRs compared to the ECRB and WCRB. Moreover, the trade-off between positioning and communication is revealed via changing critical parameters by simulations.

Explicit bounds for joint range and direction-of-arrival estimation in MIMO radar

Bayesian bounds, such as the Ziv–Zakai bound (ZZB), can provide a tight baseline for the performance of estimators or an accurate prediction for the performance limit of radar systems under all signal-to-noise ratio (SNR) conditions. However, there exists very limited literature that deals with the explicit bound for joint estimation of multi-parameters in radar systems. In this paper, we employ the thought of Shannon’s information theory to propose a tight bound with closed-form expression for joint range and direction-of-arrival (DOA) estimation in multiple-input multiple-output radar. Based on the joint a posteriori probability density function, we define range-DOA entropy error (RDEE) as the entropy power of range and DOA. With normalized a posteriori entropies in low and high SNRs, the closed-form approximation for RDEE is derived. The explicit RDEE is comprehensive and captures the effect of the SNR, the number of array elements, and the ratio of the signal bandwidth to the carrier frequency. RDEE is compared with the ZZB by a numerical simulation and the result shows RDEE and ZZB are almost identical in low and high SNRs.

Ziv-Zakai Bound for DOAs Estimation

Lower bounds on the mean square error (MSE) play an important role in evaluating the direction-of-arrival (DOA) estimation performance. Among numerous bounds for DOA estimation, the local Cram <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathrm{\acute{e}}$</tex-math></inline-formula> r-Rao bound (CRB) is only tight asymptotically. By contrast, the existing global tight Ziv-Zakai bound (ZZB) is appropriate for evaluating the single source estimation only. In this paper, we derive an explicit ZZB applicable for evaluating hybrid coherent/incoherent multiple sources DOA estimation. It is first shown that, a straightforward generalization of ZZB from single source estimation to multiple sources estimation cannot keep the bound valid in the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a priori</i> performance region. To derive a global tight ZZB, we then introduce order statistics to describe the change of the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a priori</i> distribution of DOAs caused by ordering process during the MSE calculation. The derived ZZB is for the first time formulated as a function of coherent coefficients between coherent sources, and reveals the relationship between the MSE convergency in the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a priori</i> performance region and the number of sources. Moreover, the derived ZZB also provides a unified tight bound for both overdetermined and underdetermined DOA estimation. Simulation results demonstrate the obvious advantages of the derived ZZB over the CRB on evaluating and predicting the estimation performance for multiple sources DOA.

Explicit Performance Limit for Joint Range and Direction of Arrival Estimation in Phased-Array Radar Sensors

For radar sensors in vehicle safety systems, it is of importance to establish a fundamental performance limit for the estimation accuracy of targets. Bayesian bounds, such as the ZivZakai bound (ZZB), can provide an accurate prediction for the performance limit under all signal-to-noise ratio (SNR) conditions. However, Bayesian bounds for the joint estimation of multiple parameters in radar systems have not been sufficiently studied and the explicit expression is still unavailable. In this paper, we employ the thought of Shannon's information theory to propose a tight bound with closed-form expression for joint range and direction of arrival (DOA) estimation in phased-array radar. Based on the joint <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a posteriori</i> probability density function, we define range-DOA entropy error (RDEE) as the entropy power of range and DOA. With normalized <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a posteriori</i> entropies in low and high SNRs, the closed-form approximation for the theoretical RDEE is derived. The explicit theoretical RDEE is comprehensive and captures the effect of the SNR, the number of elements, and the ratio of the signal bandwidth to the carrier frequency. The explicit theoretical RDEE is compared with the Cramér-Rao bound and the ZZB by a numerical simulation and the result shows that the theoretical RDEE can provide a more accurate performance prediction for the joint range-DOA estimation in phased-array radar.

Ziv-Zakai Bound for Compressive Time Delay Estimation

Compressive radar receiver can keep a good balance between sub-Nyquist sampling and high resolution. To evaluate the performance of compressive time delay estimators, Cram&#x00E9;r-Rao bound (CRB) has been derived for lower bounding the mean square error (MSE), which, unfortunately, is a local bound being tight in the asymptotic region only. In this paper, we use the Ziv-Zakai bound (ZZB) methodology to develop a Bayesian MSE bound on compressive time delay estimation by incorporating the <i>a priori</i> information of the unknown time delay. Specifically, we respectively derive deterministic ZZB and stochastic ZZB as functions of compressive sensing (CS) kernel, where there is no restriction on CS kernels and Gaussian noise colors. Simulation results demonstrate that compared with Bayesian CRB, ZZB provides a better performance prediction for minimum MSE estimator of compressive time delay estimation over a wide range of signal-to-noise ratios, where different CS kernels, compression ratios, <i>a priori</i> distributions and Gaussian noise colors are tested.

Arrival-time picking uncertainty: Theoretical estimations and their application to microseismic data

Traveltime-based methods depend on the accurate determination of the arrival times of seismic waves. They further benefit from information on the uncertainty with which the arrival times are determined. Among other applications, arrival-time uncertainties are used to weight data in inversion algorithms and to define the resolution of reconstructed velocity models. The most physically meaningful approaches for the estimation of arrival-time uncertainties are based on probabilistic formulations. The two approaches for the assessment of the lower bound of arrival-time uncertainties, the Cramér–Rao Bound (CRB) and the Ziv–Zakai Bound (ZZB), have been reviewed. The CRB determines the minimum-achievable estimation error under the assumption of a high signal-to-noise ratio (S/N) but underestimates said error for small S/N. The ZZB provides a better result for noisy data because it utilizes a priori information. The CRB and ZZB require knowledge of the spectral variance of the signal, which often is hard to determine in seismic experiments. Furthermore, both bounds assume additive white Gaussian noise (AWGN), which does not hold for seismic data. To overcome these problems, alternative expressions have been proposed, which yield comparable estimates as CRB and ZZB but are solely based on the S/N and the dominant period in the data. Moreover, a recipe to correct the S/N and account for the difference between the seismic noise and AWGN has been provided. For a case study of downhole microseismic monitoring, it is determined that the new expressions provide station-dependent arrival-time uncertainties, which are used as weights to improve source location uncertainties.

Ziv‐Zakai Bound for Direct Position Estimation

Direct position estimation (DPE) has arisen as an appealing alternative to the conventional two-step positioning (2SP) approach. The usual metric to assess the performance of estimators is the Cramér-Rao Bound (CRB). However, the CRB is only accurate in the high signal-to-noise ratio (SNR) region. In the lower SNR regions, high estimation errors take place and the variance of the estimates approach the a priori domain of the parameter. The DPE approach is expected to overcome the 2SP approach in such conditions. With the objective of throwing light on this aspect, this paper shows the derivation of an approximate Ziv-Zakai Bound (ZZB) for the DPE approach and a mean square error (MSE) performance approximation based on the ZZB for the conventional approach. The derived bounds are compared with their associated CRBs and also with the performance of the corresponding maximum likelihood estimator (MLE). © 2018 Institute of Navigation.

Improved Lower Bounds for Ranging in Synchronous Visible Light Positioning Systems

In this study, the Ziv-Zakai bound (ZZB) is derived for synchronous visible light positioning (VLP) systems. The proposed ZZB extracts ranging information from the prior information, the time delay parameter, and the channel attenuation factor based on the Lambertian pattern. In addition to the ZZB, the Bayesian Cramer-Rao bound (CRB) and the weighted CRB (WCRB) are calculated for synchronous VLP systems. Furthermore, a closed-form expression is obtained for the expectation of the conditional CRB (ECRB). Numerical examples are presented to compare the bounds against each other and against the maximum a posteriori probability (MAP) estimator. It is observed that the ZZB can provide a reasonable lower limit on the performance of MAP estimators. On the other hand, the WCRB and the ECRB converge to the ZZB in regions of low and high source optical powers, respectively; however, they are not tight in other regions.

Fundamental Limits on RSS Based Range Estimation in Visible Light Positioning Systems

In this study, theoretical limits are obtained for the accuracy of range (distance) estimation in visible light positioning (VLP) systems. In particular, the Ziv-Zakai bound (ZZB) and the weighted Cramer-Rao bound (WCRB) are derived for range estimation based on received signal strength (RSS) measurements. Also, the maximum a posteriori probability (MAP) and the minimum mean-squared error (MMSE) estimators are obtained for RSS based range estimation, and compared against the theoretical limits.

Ziv–Zakai Bound for Joint Parameter Estimation in MIMO Radar Systems

Local bounds, such as the Cramer-Rao bound (CRB), provide inaccurate predictions under low signal-to-noise ratio (SNR) conditions. Global bounds are capable of providing more accurate predictions of the performance of estimators over the full range of SNR. In this paper, we derive the Ziv-Zakai bound (ZZB) for joint location and velocity estimation of a target illuminated by a non-coherent multiple-input multiple-output (MIMO) radar employing orthogonal waveforms and widely spaced antennas. The setup captures the multistatic nature of the target gains, where each pair of transmit-receive elements experience independent gains governed by the Swerling type 1 model. The target returns are observed in the presence of spatially and temporally independent Gaussian clutter-plus-noise. The ZZB for joint delay and Doppler estimation for single-input and single-output (SISO) radar is also developed. We show that the ZZB is a comprehensive metric that captures the effect of the SNR, the ambiguity function (AF) and other parameters of the radar systems. The effects of different system configurations are explored through numerical studies. The results are useful for the analysis of both active and passive radars.

Matched-Field Processing Performance Under the Stochastic and Deterministic Signal Models

Matched-field processing (MFP) is commonly used in underwater acoustics to estimate source position and/or oceanic environmental parameters. Performance prediction of the multisnapshot and multifrequency MFP problem is of critical importance. To this end, two signal models are usually considered: the stochastic model, which assumes that the source signal is a stochastic process, and the deterministic model, which assumes that the source signal is a deterministic quantity. The Ziv-Zakai bound (ZZB) and the method of interval errors (MIE), which both rely on the computation of a so-called pairwise error probability, proved to be useful tools for MFP performance prediction. However, only the stochastic model has been considered so far. This paper provides a method that allows to compute the pairwise error probability, hence to use the ZZB and MIE, under both the stochastic and deterministic signal models. The proposed approach, based on recent results on quadratic forms in Gaussian variables, unifies the two models under the same formalism. The results are illustrated through the computation of the ZZB and MIE performance analysis. The Bayesian and the hybrid Cramer-Rao bounds are also given for comparison.

Compressive Parameter Estimation in AWGN

Compressed sensing is by now well-established as an effective tool for extracting sparsely distributed information, where sparsity is a discrete concept, referring to the number of dominant nonzero signal components in some basis for the signal space. In this paper, we establish a framework for estimation of continuous-valued parameters based on compressive measurements on a signal corrupted by additive white Gaussian noise (AWGN). While standard compressed sensing based on naive discretization has been shown to suffer from performance loss due to basis mismatch, we demonstrate that this is not an inherent property of compressive measurements. Our contributions are summarized as follows: (a) We identify the isometries required to preserve fundamental estimation-theoretic quantities such as the Ziv-Zakai bound (ZZB) and the Cramer-Rao bound (CRB). Under such isometries, compressive projections can be interpreted simply as a reduction in "effective SNR." (b) We show that the threshold behavior of the ZZB provides a criterion for determining the minimum number of measurements for "accurate" parameter estimation. (c) We provide detailed computations of the number of measurements needed for the isometries in (a) to hold for the problem of frequency estimation in a mixture of sinusoids. We show via simulations that the design criterion in (b) is accurate for estimating the frequency of a single sinusoid.

Performance Analysis and Design of Position-Encoded Microsphere Arrays Using the Ziv-Zakai Bound

Position-encoded microsphere arrays are a promising technology for identifying biological targets and quantifying their concentrations. In this paper we analyze the statistical performance of these arrays in imaging targets at typical low signal-to-noise ratio (SNR) levels. We compute the Ziv-Zakai bound (ZZB) on the errors in estimating the unknown parameters, including the target concentrations. We find the SNR level below which the ZZB provides a more accurate prediction of the error than the posterior Cramér-Rao bound (PCRB), through numerical examples. We further apply the ZZB to select the optimal design parameters of the microsphere array device and investigate the effects of the experimental variables such as microscope point-spread function. An imaging experiment on microspheres with protein targets verifies the optimal design parameters using the ZZB.

Lower Bounds to the Performance of Bit Synchronization for Bandwidth Efficient Pulse-Shaping

Bandwidth efficient signaling requires using pulse-shaping that inherently has intersymbol interference (ISI) except when the timing is perfect. Several lower bounds to the mean-square-error (MSE) of non-data-aided bit-synchronizers are derived for this case. The optimal lower bound is derived using the maximum likelihood (ML) criterion for a sequence of binary pulse amplitude modulated pulses in the presence of ISI and Gaussian noise. This lower bound is used as a benchmark to evaluate the performance of other synchronizers in a practical scenario. It is shown that a previous lower bound based on the ISI-free ML synchronizer cannot be used to lower bound the MSE of bit-synchronizers. A detection theory bound (DTB) (also called Ziv-Zakai bound) is applied to the symbol timing recovery problem in the presence of ISI and it is shown that this bound is a tight lower bound on the MSE of the ML synchronizer. A simple lower bound on this DTB is derived and it is shown that the simple bound is almost as tight as the well known modified Cramer-Rao bound (MCRB) at moderate values of SNR, while it does not suffer from the shortcomings of the MCRB at small values of SNR.

Ziv–Zakai Bounds on Time Delay Estimation in Unknown Convolutive Random Channels

Using the Ziv-Zakai bound (ZZB) methodology, we develop a Bayesian mean-square error bound on time delay estimation (TDE) in convolutive random channels, and compare it with time delay estimator performance and a Crame¿r-Rao bound. The channel is modeled as a tapped delay line, whose taps are Gaussian random variables that may be nonzero mean and correlated, a model widely adopted in many applications such as wideband fading in a multipath channel. The time delay has a uniform prior distribution. A ZZB is developed that incorporates the prior on the random time delay, as well as the convolutive random Gaussian channel, and does not assume the receiver has knowledge of the channel realization. The ZZB provides good performance prediction for maximum a posteriori (MAP) time delay estimation, tracking the low, medium, and high signal-to-noise ratio (SNR) regimes. The convolutive channel model includes important special cases, such as narrowband Gaussian channels corresponding to Rayleigh/Rician fading, wideband multipath channels with a power decay profile (such as exponential decay), and known channels. We show that the associated Crame¿r-Rao bound is tight only at high SNR, whereas the ZZB predicts threshold behavior and TDE breakdown as the SNR decreases. When compared to a ZZB that assumes knowledge of the channel realization, the ZZB developed here provides a more realistic and tighter bound, revealing the performance loss due to lack of channel knowledge. The MAP estimator incorporates knowledge of the channel statistics, and so performs much better than a maximum likelihood estimator that minimizes mean square error but does not use knowledge of the random channel statistics. Several examples illustrate the estimator and bound behaviors.

Time Delay Estimation Bounds in Convolutive Random Channels

We develop bounds on time delay estimation (TDE) in wideband convolutive random channels. The improved Ziv-Zakai bound (ZZB) is adapted to this case, resulting in mean-square error (MSE) bounds that are independent of estimator bias, and avoid regularity conditions. We assume a known transmitted signal, a tapped delay line random channel model, and a uniform prior on the delay. The taps are modeled as Gaussian, including non-zero mean and possible correlation between taps. The channel model may be tuned to a variety of environments through the choice of mean and covariance, and models wideband and ultra-wideband line of sight (LOS) and non-LOS (NLOS) cases. Flat fading is a simple special case with one tap. The channel power delay profile is incorporated through the tap covariance matrix, and results for an exponential power delay profile are given. We derive the ZZB for a given channel realization, and then average over the random channel model. The resulting bound reflects the impact of the random channel, providing an average MSE performance bound, that assumes perfect channel estimates are available to the receiver. Specific cases and examples illustrate the impact of the key system, channel, and signal parameters.

Bayesian Bounds for Matched-Field Parameter Estimation

Matched-field methods concern estimation of source locations and/or ocean environmental parameters by exploiting full wave modeling of acoustic waveguide propagation. Because of the nonlinear parameter-dependence of the signal field, the estimate is subject to ambiguities and the sidelobe contribution often dominates the estimation error below a threshold signal-to-noise ratio (SNR). To study the matched-field performance, three Bayesian lower bounds on mean-square error are developed: the Bayesian Crame/spl acute/r-Rao bound (BCRB), the Weiss-Weinstein bound (WWB), and the Ziv-Zakai bound (ZZB). Particularly, for a multiple-frequency, multiple-snapshot random signal model, a closed-form minimum probability of error associated with the likelihood ratio test is derived, which facilitates error analysis in a wide scope of applications. Analysis and example simulations demonstrate that 1) unlike the local CRB, the BCRB is not achieved by the maximum likelihood estimate (MLE) even at high SNR if the local performance is not uniform across the prior parameter space; 2) the ZZB gives the closest MLE performance prediction at most SNR levels of practical interest; 3) the ZZB can also be used to determine the necessary number of independent snapshots achieving the asymptotic performance of the MLE at a given SNR; 4) incoherent frequency averaging, which is a popular multitone processing approach, reduces the peak sidelobe error but may not improve the overall performance due to the increased ambiguity baseline; and finally, 5) effects of adding additional parameters (e.g., environmental uncertainty) can be well predicted from the parameter coupling.

Bayesian bounds on parameter estimation accuracy for compact coalescing binary gravitational wave signals

A global network of laser interferometric gravitational wave detectors is projected to be in operation by around the turn of the century. Here, the noisy output of a single instrument is examined. A gravitational wave is assumed to have been detected in the data and we deal with the subsequent problem of parameter estimation. Specifically, we investigate theoretical lower bounds on the minimum mean-square errors associated with measuring the parameters of the inspiral waveform generated by an orbiting system of neutron stars/black holes. Three theoretical lower bounds on parameter estimation accuracy are considered: the Cramer-Rao bound (CRB); the Weiss-Weinstein bound (WWB); and the Ziv-Zakai bound (ZZB). We obtain the WWB and ZZB for the Newtonian-form of the coalescing binary waveform, and compare them with published CRB and numerical Monte-Carlo results. At large SNR, we find that the theoretical bounds are all identical and are attained by the Monte-Carlo results. As SNR gradually drops below 10, the WWB and ZZB are both found to provide increasingly tighter lower bounds than the CRB. However, at these levels of moderate SNR, there is a significant departure between all the bounds and the numerical Monte-Carlo results.